Modified Black-Scholes Model Research Articles

Application of Lie symmetries to solving modified Black-Scholes equation

We perform Lie symmetry analysis on the modified Black-Scholes model described as a partial differential equation (PDE). As a result, a new complete Lie symmetry group and infinitesimal generators of the one-dimensional modified fractional Black-Scholes model are derived. Furthermore, we compute a family of exact invariant solutions that constitute the modified fractional Black-Scholes model using the associated infinitesimal generators and the corresponding similarity reduction equations. Using known solutions, more solutions are generated via group point transformations.

Incorporating the impacts of climate change in transportation infrastructure decision models

Abstract Historically an important transportation corridor in the Northwest Territories, climate change has shortened the duration of the Mackenzie River’s navigational season. Communities rely increasingly on airlift as the growing volatility affects barging operations, leading to higher overall freight costs. Using an options approach, we present a methodological framework that supports flexible infrastructure decision making, accounting for the impacts of climate change uncertainty. We apply this method to the decision of whether to continue barging on the Mackenzie River, or connect the entire corridor by extending the all-weather Mackenzie Valley Highway, explicitly considering uncertainties in river barging conditions. We first model river open season days as a stochastic process; barging is dependent on the number of open season days, which in turn is affected by climate change. Second, we evaluate the expected cost of barging and airlift each season using a modified Black-Scholes model. Finally, we use real options to determine how long construction of the all-weather highway may be deferred. The results indicate that it is advisable to defer construction nearly a decade, in balancing the costs of construction against climate change uncertainty. This paper demonstrates that when we explicitly incorporate the impact of climate change on project valuations, particularly those in northern and Arctic Canada where these impacts are considerable, project valuations can change significantly such that all-weather road construction is supported, even if it is deferred to future years. This method can assist federal and territorial governments in communicating the impacts of climate change on communities, and provide another tool to support multi-layered, complex transportation infrastructure investment decisions that address these rapidly changing environments.

Algorithm for Determining the Volatility Function in the Black–Scholes Model

An algorithm for reconstructing the volatility function in the modified Black–Scholes model is developed. Results of numerical computations are presented. It is shown that adding information about the prices of similar options with different issue dates makes it possible to improve the accuracy and increase the interval in which the volatility function can be reconstructed.

Empirical Performance of the Black-Scholes Model in India

In this paper we empirically investigate the performance of the Black-Scholes (BS) model and its skewness-kurtosis adjusted modified version in Indian stock market using data on Nifty index call & put options over the period July, 2007 to June 2009. The presence of volatility smile in index options indicates that benchmark BS model is mis-specified. We also find that the pricing performance of the modified BS model is not significantly better than that of the benchmark BS model. In addition to the BS model, nine specifications of the ad-hoc Black-Scholes (AHBS) model are also examined. Among the nine model specifications examined only the model with standardized moneyness as independent variable best describes the smile pattern. The mis-pricing (measured in terms of MAPE) is highest in out-of-the-money and near maturity call and put options, while mis-pricing is lowest for the model with standardized moneyness as independent variable.

Effectiveness of the Skewness and Kurtosis Adjusted Black-Scholes Model in Pricing Nifty Call Options

This paper tests the predictive accuracy of the Black-Scholes (BS) model in pricing the Nifty index option contracts and examines whether the skewness and kurtosis adjusted BS model of Corrado and Su (1996) gives better results than the original BS model. We also examine whether volatility smile in case of NSE Nifty options, if any, can be attributed to the non normal skewness and kurtosis of stock returns. We use S&P CNX NIFTY near-the-month call options for the period January 1, 2003 to December 24, 2008. The results show that BS model is misspecified as the implied volatility graph depicts the shape of a ‘Smile’ for the study period. There is significant underpricing by the original BS model and that the mispricing increases as the moneyness increases. Even the modified BS model misprices options significantly. However, pricing errors are less in case of the modified BS model than in case of the original BS model. On the basis of Mean Absolute Error (MAE) it can be concluded that the modified BS model is performing better than the original BS model. Moreover, volatility smile in case of NSE Nifty options for the study period cannot be attributed to the non normal skewness and kurtosis of stock returns.

Empirical performance of alternative pricing models of currency options

This article examines the out-of-sample pricing performance and biases of the Heston’s stochastic volatility and modified Black-Scholes option pricing models in valuing European currency call options written on British pound. The modified Black-Scholes model with daily-revised implied volatilities performs as well as the stochastic volatility model in the aggregate sample. Both models provide close and similar correspondence to actual prices for options trading near- or at-the-money. The prices generated from the stochastic volatility model are subject to fewer and weaker aggregate pricing biases than are the prices from the modified Black-Scholes model. Thus, the stochastic volatility model may provide improved estimates of the measures of option price sensitivities to key option parameters that may lead to more effective hedging and speculative strategies using currency options. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:265–291, 2000

Empirical performance of alternative pricing models of currency options

This article examines the out-of-sample pricing performance and biases of the Heston’s stochastic volatility and modified Black-Scholes option pricing models in valuing European currency call options written on British pound. The modified Black-Scholes model with daily-revised implied volatilities performs as well as the stochastic volatility model in the aggregate sample. Both models provide close and similar correspondence to actual prices for options trading near- or at-the-money. The prices generated from the stochastic volatility model are subject to fewer and weaker aggregate pricing biases than are the prices from the modified Black-Scholes model. Thus, the stochastic volatility model may provide improved estimates of the measures of option price sensitivities to key option parameters that may lead to more effective hedging and speculative strategies using currency options. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:265–291, 2000

Estimation of empirical pricing equations for foreign-currency options: Econometric models vs. arbitrage-free models

This paper reviews the econometric models used in estimating option or warrant prices and finds most of them violating arbitrage-free conditions. However, we find the indirect form proposed by Kassouf (1965) a natural choice for call-option pricing model. By proving six mathematical properties, we show that the Kassouf's form not only satisfy the major boundary conditions for call options but also is flexible enough to span all feasible option prices. The Kassouf model is modified to estimate empirical pricing equation for foreign-currency options. Empirical results show that the modified Kassouf model performs the best among the econometric models. The Kassouf model is tested against two Black-Scholes-type arbitragefree models for foreign-currency options. Preliminary in-sample test show that the modified Kassouf model performs better than the modified Black-Scholes model and also outperforms the mixed-jump option-pricing model using grouped data.

Pricing European Currency Options: A Comparison of the Modified Black- Scholes Model and a Random Variance Model

We use the modified Black-Scholes model and a random variance option pricing model to study prices of European currency options traded in Geneva. The options, which cannot be exercised early, include calls and puts on the dollar/Swiss franc exchange rate. In the empirical analysis, we examine the model fit and the biases with respect to the strike price, time to maturity, and volatility. There is some evidence of mispricing and there are small gains available by trading with the random variance model.